The present invention, in some embodiments thereof, relates to Computed Tomography (CT), and, more particularly, but not exclusively, to methods for reconstruction of images from projection data in computed tomography.
A Computed Tomography (CT) scan enables estimation of attenuation coefficients of scanned object components by comparing a flow of photons entering and exiting the scanned object along straight lines. Theoretically, a log-transformed photon count information corresponds to an X-ray transform of the attenuation function, and should provide a perfect reconstruction. In practice, measurement data is limited by a discrete sampling scheme, and is degraded by a number of physical phenomena occurring in a scanner. The first mentioned problem is an inevitable but minor cause for the limited resolution in the CT images; mainly, the images are corrupted by the data degradation factors. Some of the factors are: off-focal radiation; detector afterglow and crosstalk; beam hardening; and Compton scattering. The factors introduce a structured bias into the measurements.
Another source of deterioration, which is dominant in a low-dose scenario, is stochastic noise. One type of noise stems from low photon counts, which occur when the X-rays pass through high-attenuation areas. The phenomenon is similar to the photon starvation occurring in photo cameras in poor lighting conditions. Statistically, in such cases, data is modeled as an instance of Poisson random variables.
Another type of noise originates from the noise present in the detectors. This noise is modeled as an additive Gaussian random process.
A basic reconstruction transform, Filtered Back-Projection (FBP) [3], takes a limited account of the noise statistics: FBP employs a low-pass 1-D convolution filter in the projection domain, which parameters are preset for specific anatomical regions and standard scan protocols. As a result, the problem of photon starvation manifests in the output image in the form of streak artifacts. Each measured line integral is effectively smeared back over that line through the image by the back-projection; an incorrect measurement results in a line of wrong intensity in the image. Typically, the streaks radiate from bone regions or metal implants, which corrupt its contents and jeopardize its diagnostic value.
Images of better quality—having reduced artifacts and increased spatial resolution—are obtained with statistically based methods, which solve the Maximum-a-Posteriori (MAP) problem. In this case the MAP problem is expressed as a minimization of the Penalized Likelihood (PL) objective function. The likelihood expression models the aforementioned physical phenomena associated with the scan as well as the noise statistics, and an additional penalty component models expected properties of the CT images, that is, includes prior information about an image to be reconstructed. The PL objective can be designed to restore a true sinogram from noisy observations [4], [5] or to reconstruct an output CT image [6]. Usually, the PL equation is difficult to solve, so it is replaced by the second-order approximation, Penalized Weighted Least Squares (PWLS) [6], [7]. A drawback of reconstruction based on explicit statistical modeling is a computationally heavy iterative solution.
An alternative approach to the problem is to use adaptive signal processing techniques, implicitly modeling the signal properties. The techniques are applied in a non-iterative fashion, and have a computational complexity comparable to the FBP. Demirkaya [8] uses a nonlinear anisotropic diffusion filter for sinogram de-noising to reduce streak artifacts. For the same purpose, Hsieh [9] employs a trimmed mean filter adaptive to the noise variance. For each detector reading x, the algorithm adaptively chooses a number of its neighbors participating in the filtering operation and then the value of x is replaced with the trimmed mean of these neighbors (a portion of highest and lowest values among the neighbors is discarded). Thus, to some extent, the aforementioned statistical model of the scan is used: noisy samples get stronger filtering than the more reliable ones. Experimental results in the above-mentioned work are impressive.
A similar concept, with a different kind of filter, is adopted by the work of Kachereiss et al, who apply adaptive convolution-based filtering in the projection domain [10]. The filter width is data dependent, so that their algorithm leaves low-valued sinogram elements untouched.
Beyond the use of general-purpose tools, algorithms are known which apply machine learning methods to perform the processing, adaptive to tomographic data.
An example algorithm is described in [11]: measured projections are locally filtered according to a preliminary classification of regions in the measured projections. Classes and corresponding filters are derived automatically, via an off-line exemplar-based training process. The standard smoothing by a low-pass convolution filter is replaced with locally-adaptive filtering, optimized for signal quality on the training set. A common property of those listed works is a fast, non-iterative sinogram processing, which take a limited account of the statistical model for the measurements.
Finally, there are also post-processing methods operating in the image domain. Sauer and Liu design a set of non-stationary filters, trying to mend the effect of low-count noise [12]. Wavelet-based method is employed in the work of Borsdorf et al [13]. Using the recently available dual-source CT scan, the algorithm builds two versions of the reconstructed image from disjoint sets of measurements, and exploits a correlation between the wavelet coefficients of the two versions in order to reduce image noise.
Additional background art includes:    [1] J. Bian P. J. La Riviere and P. A. Vargas, “Penalized-likelihood sinogram restoration for computed tomography” IEEE Trans. Med. Imag., vol. 25, no. 8, pp. 1022-36, August 2006.    [2] K. D. Sauer J. Hsieh J.-B. Thibault, C. A. Bouman, “A recursive filter for noise reduction in statistical iterative tomographic imaging” in SPIE/IS&T Conference on Computational Imaging IV. 2006, vol. 6065, pp. 60650X-60650X-10, SPIE.    [3] F. Natterer, F, Wubbeling, Mathematical methods in image reconstruction, Section 5.1, in SIAM, 2001.    [4] J. Bian P. J. La Rivirre and P. A. Vargas, “Comparison of quadratic and median-based roughness penalties for penalized-likelihood sinogram restoration in computed tomography” Int. J. Biomed. Imaging, vol. 2006, pp. 1-7, 2006.    [5] J. Wang J. Wen H. Lu J. Hsieh T. Li, X. Li and Z. Liang, “Nonlinear sinogram smoothing for low-dose X-ray CT” IEEE Trans. Nucl. Sci., vol. 51, no. 5, pp. 2505-2513, 2004.    [6] I. A. Elbakri and J. A. Fessler, “Statistical image reconstruction for polyenergetic X-ray computed tomography” IEEE Trans. Med. Imag., vol. 21, no. 2, pp. 89-99, February 2002.    [7] H. Lu Z. Liang J. Wang, T. Li, “Penalized weighted least-squares approach to sinogram noise reduction and image reconstruction for low dose X-ray computed tomography” IEEE Trans. Med. Imag., vol. 25, no. 10, pp. 1272-1283, 2006.    [8] O. Demirkaya, “Reduction of noise and image artifacts in computed tomography by nonlinear filtration of projection images” 2001, vol. 4322, pp. 917-923, SPIE.    [9] J. Hsieh, “Adaptive streak artifact reduction in computed tomography resulting from excessive X-ray photon noise” Medical Physics, vol. 25, no. 11, pp. 2139-2147, 1998.    [10] O. R. Watzke M. Kachelriess and Willi A. Kalender, “Generalized multidimensional adaptive filtering for conventional and spiral single-slice, multi-slice, and cone-beam CT” Medical Physics, vol. 28, no. 4, pp. 475-490, 2001.    [11] K. D. Sauer B. I. Andia and C. A. Bouman, “Nonlinear backprojection for tomographic reconstruction” IEEE Trans Nucl. Sci., vol. 49, no. 1, pp. 61-68, February 2002.    [12] K. D. Sauer and B. Liu, “Nonstationary filtering of transmission tomograms in high photon counting noise” IEEE Trans. Med. Imag., vol. 10, no. 3, pp. 445-452, 1991.    [13] T. Flohr A. Borsdorf, R. Raupach and J. Hornegger, “Wavelet based noise reduction in CT-images using correlation analysis,” IEEE Trans. Med. Imag., vol. 27, no. 12, pp. 1685-1703, 2008.    [14] Ing-Tsung Hsiao H. Lu, X. Li and Zhengrong Liang, “Analytical noise treatment for low-dose CT projection data by penalized weighted least square smoothing in the k-l domain” Proceedings of SPIE.    [15] Y. Hel-Or and D. Shaked, “A discriminative approach for wavelet denoising” IEEE Trans. Im. Proc., vol. 17, no. 4, 2008.    [16] G. N. Ramachandran and A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: Application of convolutions instead of Fourier transforms” Proceedings of the National Academy of Sciences of the United States of America, vol. 68, no. 9, pp. 2236-2240, 1971.    [17] D. L. Donoho A. M. Brookstein and M. Elad, “From sparse solutions of systems of equations to modeling of signals and images” SIAM Review, vol. 51, no. 1, pp. 34-81, 2009.    [18] J. Shtok M. Zibulevsky M. Elad, B. Matalon, “A wide-angle view at iterated shrinkage algorithms” in SPIE (Wavelet XII), 2007, pp. 26-29.    [19] M. Zibulevsky and M. Elad, “L1-12 optimization in signal and image processing” IEEE Signal Processing Magazine, vol. 27, no. 3, pp. 78-88, 2010.    [20] M. Elad, “Why simple shrinkage is still relevant for redundant representations?” IEEE Trans. on Information Theory, vol. 52, no. 12, pp. 5559-5569, December 2006.    [21] D. L. Donoho and I. M. Johnston, “Ideal spatial adaptation via wavelet shrinkage” Biometrika, vol. 81, no. 3, pp. 425-455, 1994.    [22] Y. Hel-Or A. Adler and M. Elad, “A shrinkage learning approach for single image super-resolution with overcomplete representations” ECCV 2010, pp. 622-635, 2010.    [23] F. J. Anscombe, “The transformation of Poisson, binomial and negative binomial data” Biometrika, vol. 35, no. 3/4, pp. 246-254, 1948.    [24] K. I. Kim J. H. Kim and C. E. Kwark, “A filter design for optimization of lesion detection in spect” IEEE Nuclear Science Symposium, vol. 3, pp. 1683-1687, 1996.    [25] J. Nocedal and S. J. Wright, Numerical Optimization, Section 6.1, Springer-Verlag, second edition, 2006.    [26] J. A. Fessler and W. L. Rogers, “Spatial resolution properties of penalized-likelihood image reconstruction: space-invariant tomographs”, IEEE Trans. Im. Proc., vol. 5, no. 9, pp. 1346-1358, 1996.    [27] N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete Cosine Transform”, IEEE Trans. Computers, pp. 90-93, 1974.
The disclosures of all references mentioned above and throughout the present specification, as well as the disclosures of all references mentioned in those references, are hereby incorporated herein by reference.